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Modeling and Computation of Mean Field Equilibria in Producers' Game with Emission Permits Trading

This paper proposes a mean field game model incorporating emission permits trading to analyze producers' behaviors and develops a fitted finite volume method to solve the resulting coupled HJB and Kolmogorov equations, demonstrating that higher permit prices incentivize producers to adopt lower emission levels.

Original authors: Shuhua Chang, Xinyu Wang, Alexander Shananin

Published 2026-06-04

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CC BY 3.0

Original authors: Shuhua Chang, Xinyu Wang, Alexander Shananin

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive factory floor with thousands of identical workers. Each worker is trying to make as much money as possible, but they are all competing for the same customers. If everyone tries to produce at the exact same level, they undercut each other, and everyone makes less money. This is what economists call "negative externality"—your success hurts my success if we do the same thing.

Now, add a twist: the government is watching their smokestacks. There is a limit on how much pollution (carbon emissions) is allowed. But instead of just saying "stop," the government creates a market for pollution permits.

If you pollute less than your allowance, you can sell your extra permits for cash.
If you pollute more than your allowance, you must buy permits from others.

This paper is a mathematical story about how these thousands of workers behave when they realize they can trade these pollution permits.

The "Mean Field" Concept: The Ocean vs. The Drop

In normal game theory, you might study how 5 or 10 people play a game. But here, there are so many workers that no single worker matters on their own. It’s like a drop of water in the ocean. One drop doesn’t change the ocean’s level, but the ocean’s level determines how the drop behaves.

This is called Mean Field Game theory. Instead of tracking every single worker, the mathematicians track the "average crowd." They ask: If I am one worker, and I see what the average crowd is doing, what is the best move for me? And then they ask: If everyone thinks like that, what does the crowd look like?

The Mathematical Engine: Two Dancing Equations

To solve this, the authors use two complex mathematical equations that dance together:

The HJB Equation (The Planner): This looks backward in time. It helps each worker figure out, "If I want to maximize my profit by the end of the year, what should I do right now?" It’s like a GPS calculating the best route based on the destination.
The Kolmogorov Equation (The Crowd): This looks forward in time. It tracks how the crowd of workers moves around. If many workers decide to lower their pollution, this equation shows the crowd shifting toward the "low pollution" side of the graph.

These two equations are linked. The crowd’s behavior affects the worker’s plan, and the worker’s plan affects the crowd’s behavior. It’s a loop.

The Solution: A "Fitted" Puzzle

Solving these equations is hard because they are messy and continuous. The authors invented a numerical method called the "Fitted Finite Volume Method."

Think of it like solving a jigsaw puzzle where the pieces are slightly warped. Instead of forcing square pegs into round holes (which is what older methods did), this method "fits" the mathematical pieces to the actual shape of the problem. It breaks the problem into tiny chunks and solves them one by one, ensuring the math stays stable and accurate. The authors proved that their method is reliable and converges to the correct answer quickly.

What Did They Find?

They ran computer simulations to see what happens under different conditions. Here are the key takeaways, explained simply:

The "Don’t Be Average" Effect:
At the start, imagine most workers are producing at a medium pollution level. But because of the competition (negative externality), if you’re in the middle, you’re fighting too many rivals. So, workers naturally spread out. Some go to very low pollution, some to very high. The crowd disperses to avoid stepping on each other’s toes.
The Price of Pollution Changes Behavior:
This is the big finding. The authors tested what happens when the price of pollution permits goes up.
- Cheap Permits: Workers don’t care much. They pollute freely.
- Expensive Permits: Suddenly, polluting costs a lot of money. Workers start shifting toward lower pollution levels. Why? Because if they pollute less, they can sell their extra permits for a nice profit.
- The Result: As the price of permits rises, the "crowd" shifts toward cleaner production. More workers choose low emission levels because it becomes financially smarter to be clean.

In a Nutshell

This paper builds a mathematical model of a huge group of producers trading pollution rights. It shows that when pollution permits are expensive, the market naturally pushes producers to be cleaner, not because they are being nice, but because it makes more money. The authors also created a new, efficient mathematical tool to calculate this behavior accurately.

Technical Summary: Modeling and Computation of Mean Field Equilibria in Producers’ Game with Emission Permits Trading

Problem Statement
This paper addresses the modeling of production behaviors for a continuum of agents (producers) whose carbon emissions are regulated by government policy, specifically within an emission permits trading scheme. The authors aim to study the productive behaviors of these agents under the background of climate change and emission reduction. In this framework, each agent is anonymous, possesses an initial quota of emission permits, and can flexibly trade permits on the market—purchasing additional permits if their quota is insufficient or selling unused permits. The core challenge is to determine the mean field equilibrium, which involves solving a coupled system of partial differential equations (PDEs) that describe the evolution of the agents' states and their optimal decision-making processes.

Methodology
The authors construct a finite-horizon mean field game model where the state variable is the emission level $E(t)$ , governed by a stochastic differential equation involving a control variable $\tau(t, E)$ representing the level of emission reduction. The dynamics include stochastic disturbances modeled by Brownian motion and reflection terms to keep emissions within bounds $[E_{min}, E_{max}]$ .

The equilibrium of the game is characterized by a coupled system consisting of:

A backward Hamilton-Jacobi-Bellman (HJB) equation satisfied by the adjoint state $v$ , which represents the value function for the optimal control problem.
A forward Kolmogorov (Fokker-Planck) equation satisfied by the density of agents $m(t, E)$ .

To solve this coupled system numerically, the authors propose a fitted finite volume method. This method combines a finite volume formulation with a fitted local approximation to the solution. Specifically:

Spatial Discretization: The domain is divided into sub-intervals. The flux at the mid-points of these intervals is approximated by solving a sequence of two-point boundary value problems analytically. This approach ensures that the resulting system matrix is an M-matrix, guaranteeing monotonicity and satisfaction of the discrete maximum principle.
Time Discretization: A two-level implicit time-stepping method is employed. The scheme allows for flexibility, reducing to the Crank-Nicolson scheme (second-order accuracy) or the backward Euler scheme (first-order accuracy) depending on the splitting parameter $\theta$ .
Algorithm: An iterative algorithm is used to resolve the coupling between the control variable $\tau$ , the density $m$ , and the adjoint state $v$ . The process involves initializing $\tau$ , solving the Kolmogorov equation for $m$ , solving the HJB equation for $v$ , and updating $\tau$ until convergence within a specified tolerance.

Key Contributions

Modeling: The paper presents a mean field game model that explicitly incorporates revenues from emission permits trading, addressing a gap in existing literature where few studies consider this specific mechanism within mean field games.
Numerical Method: The introduction of the fitted finite volume method for solving the coupled HJB and Kolmogorov equations. The authors prove that the system matrix of the discrete equations is an M-matrix, which ensures the stability and uniqueness of the numerical solution.
Analysis of Equilibrium: The study provides a computational framework to analyze how emission permit prices influence the distribution of producers' emission levels.

Results
Numerical experiments are conducted to validate the efficiency and usefulness of the proposed method.

Convergence: The method demonstrates a convergence rate of approximately $O(h^{1.9843})$ in the discrete $L^\infty$ -norm, indicating nearly second-order accuracy.
Equilibrium States:
- In a scenario where agents initially follow a normal distribution of emission levels, the density disperses over time due to negative externalities (competition), causing agents to choose different emission levels to maximize revenue.
- Effect of Permit Prices: The results show that as the price of emission permits increases, the density of agents at lower emission levels increases, while the density at higher emission levels decreases. This indicates that expensive permits incentivize producers to reduce their emissions.
- Time-Dependent Prices: In a case with time-varying permit prices, the algorithm remains stable even with non-differentiable initial conditions, and the same trend of shifting toward lower emissions with higher prices is observed.

Significance
The paper claims that its methodology, combining partial differential equations with numerical methods, contributes to solving complex problems in interdisciplinary fields. Specifically, it demonstrates that mean field games can effectively model large-scale economic systems with environmental regulations. The findings highlight the influence of emission permits trading on producers' behaviors, showing that market mechanisms (permit prices) can effectively steer producers toward lower emission levels. The authors anticipate that this approach can be applied to other complex systems arising from the intersection of economics and environmental policy.

The "Mean Field" Concept: The Ocean vs. The Drop

The Mathematical Engine: Two Dancing Equations

The Solution: A "Fitted" Puzzle

What Did They Find?

In a Nutshell

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